In this paper, we are concerned with a Boussinesq system of Benjamin–Bona–Mahony type equation, posed on a bounded interval, modeling the two-way propagation of surface waves in a uniform horizontal channel filled with an irrotational, incompressible and inviscid liquid under the influence of gravitation. The main focus is on the boundary controllability property, which corresponds to the question of whether the solutions can be driven to a given state at a given final time by means of controls acting at one endpoint of the interval. We first show that the system is not spectrally controllable. This means that no finite linear combination of eigenfunctions associated with the state equations, other than zero, can be steered to zero. Although the system is not spectrally controllable, it can be shown that it is approximately controllable, i.e., any state can be steered arbitrarily close to another state.

中文翻译：

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Boussinesq系统对色散波双向传播的可控性

在本文中，我们关注本杰明-波纳-马洪尼型方程的Boussinesq系统，该系统位于有界区间上，模拟表面波在均匀水平通道中的双向传播，该水平通道中充满了不可旋转，不可压缩且无粘性的液体，引力的影响。主要关注点是边界可控性，该属性对应于以下问题：是否可以通过作用于间隔的一个端点的控件在给定的最终时间将解决方案驱动到给定状态。我们首先显示该系统在光谱上不可控。这意味着除了零以外，与状态方程相关的本征函数的有限线性组合都不能转向零。尽管该系统不是频谱可控的，但可以证明它是近似可控的，即